Rotational correlation function

If the second term on the right-hand side of the equation is omitted, the latter is transformed into Eq. (2.76). As the omission is possible only for t < tj, Fourier transformation of the reduced equation holds for co-tj 1 only. Consequently, the equality (2.75) is of asymptotic character, and may not be utilized to find full g(co) or its Fourier-transform Kj(t) at any times. When it was nevertheless used in [117], the rotational correlation function turned out to be alternating in sign. The oscillatory behaviour of Kj(t) occured not only in a compressed gas, but also at normal pressure, when Kj(t) should vanish monotonically, if not exponentially. The origin of these non-physical oscillations is easily... 

Dill J. F., Litovitz T. A., Bucaro J. A. Molecular reorientation in liquids by Rayleigh scattering pressure dependence of rotational correlation functions, J. Chem. Phys. 62, 3839-50 (1975). 

Table 3 Average rotational correlation functions (t ) of water in various environments ...

Although MC does not have a proper time variable, the time equivalent of MC moves can be estimated, for example, by computing rotational correlation functions or molecular diffusion coefficients in well characterized liquids these are time-dependent quantities also experimentally accessible, and the comparison between the number of MC moves and the corresponding experimental data provides the required time equivalents. An order of magnitude estimate with typical MC translational or rotational steps of 0.2 A and 4° gives 1 or 2 ps per 10 translational or rotational MC moves, respectively. 

The effects of such collisionally induced kicks are treated within the so-called pressure broadening (sometimes called collisional broadening) model by modifying the free-rotation correlation function through the introduction of an exponential damping factor exp( -Itl/x) ... 

R.L.Armstrong, S.M.Blumenfeld, and C.G.Gray. Infrared spectra, rotational correlation functions and intermolecular mean squared torques in compressed gaseous methane. Can.J.Phys., 46 1331,1968. 

Consider first the case of free-rotating molecules for which the rotational correlation function is given by the expression... 

MD simulations of [C6mim][PF6] and water mixtures were carried out [98], The authors found water to be closely associated with the anions and that its presence enhances both the translational and rotational dynamics of the IL. The mean square displacement and rotational correlation functions reflected that the diffusive regime was achieved faster when water was present in the IL or the observed decay of correlations was faster when water was present, respectively [98], From this the authors deduced that the experimentally observed decrease in viscosity is a consequence of the faster translational and rotational dynamics caused by the presence of water [98], In agreement with experiments, the authors found that the fluorescence spectra of Coumarin-153 is red-shifted because of the presence of water [98],... 

This ansatz can be rationalized by some theoretical considerations [325,326]. It is also supported by the experimental data at very low concentrations of the component A where the study is reduced to the dynamics of the probe A in host B. Each probe molecule experiences the same environment, which eliminates the complications from concentration fluctuations. We have mentioned in Section III, paragraph 4, that the probe rotational correlation function indeed has the Kohlrausch form. The differential between the probe rotational time xA and the host a-relaxation time xaB is gauged by their ratio, xA/xaB. As expected, the slower the host B compared with the probe A, the larger the coupling parameter, nA = (1 — pA), obtained from the stretch exponent (3A of the measured probe correlation function. The experimental data are shown in Fig. 52. For more details, see Ref. 172. 

Thus, the combined SE and the DSE equations predict that the product Z)(T (Z)(r< )sE,DSE should equal 2r//9. Measurements of probe translational diffusion and rotational diffusion made in glass-formers have found that the product DfTc can be much larger than this value, revealing a breakdown of the SE and the DSE relations. There is an enhancement of probe translational difiusion in comparison with rotational difiusion. The time dependence of the probe rotational time correlation functions r t) are well-described by the KWW function, exp[-(t/rc)] " . The ratio A c/(A c)se,dse evaluated at T=Tg is a measure of the degree of the breakdown of the SE and DSE relations for various combinations of probes and host glass-formers. A strong correlation was observed at T=Tg between the quantity AA(A7c)se,dse and n, the width of the dispersion of the probe rotational correlation functions tit) The variation of the dispersion or n of the probe in different hosts was traced to the difference between the probe rotation time and the host stmctural relaxation time. ... 

Usually we are interested in the (auto)correlation function G(r) of an observable (i.e., a function of some stochastic coordinates). In the following we will consider either rotational correlation functions (i.e., involving the spherical harmonics Ty ,(ii])) or momentum correlation functions (i.e., involving the components of L,) for the first rotator (body 1), identified as the solute molecule... 

The matrix elements for the rotational correlation function (F = Ypq) are given by... 

In the last few years Nelson and co-workers [63-65] have presented a new approach to light scattering spectroscopy, named impulsive stimulated light scattering (ISS), which seems to be able to detect one particle rotational correlation functions. In ISS, one induces coherent vibrational motion by irradiating the sample with two femtosecond laser pulses, and... 

Diezemann and co-workers have recently presented numerical solutions of a master equation for the orientational relaxation of rotational correlation functions in glass-formers such as salol [172,173]. They have considered both globally and locally connected models, where in the latter case only states similar in energy to the starting point can be reached in one hop. They report that the main features of their results are qualitatively the same for these two extremes of connectivity. 

Using the methods of St. Pierre and Steele, Yang and Bersohn were able to evaluate the rotational correlation functions exactly for linear and spherical tops, and derive a general expression in terms of a sum of one dimensional integrals for symntetric molecular rotors. For asynunetric tops it was necessary to employ an interpolation procedure. In the case of a symmetric rotor, the sununation in (4S) reduces to three terms... 

It can be noted that, in principle, the rotational correlation functions can be measured by various molecular spectroscopies (e.g., IR, Raman, NMR) if suitable penetrants are chosen, but we are not aware of such measurements. Experimental correlation functions, together with atomistic modelling of the polymer-penetrant system, could yield important information about size and, possibly, shape distributions of the cavities inside the polymer, as probed by small molecules. 

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